Factorial (2 or more Way) ANOVA
Mikhilesh Dehane
8/1/2020
Factorial (2 or more Way) ANOVA - A factorial ANOVA is an Analysis of Variance test with more than one independent variable.
The data set from the first class contains another column that includes whether the valuation was made for a product or gamble so we have data that we can analyze using a factorial ANOVA.
For example, if we are interested in the effects of Perspective and Item Type and their potential interaction effects, we would set up our experiment so that each factor is present at each other factors levels TAble- Chooser Buyer Seller Product CP BP SP Gamble CG BG SG
Procedure - 1. Propose a research question(s) based on your questions. 2. Perform a (or more) hypotheses 3. Check the assumptions 4. Perform tests 5. Make conclusions based on the finding.
setwd("E:\\mikhilesh\\HU Sem VI ANLY 510 and 506\\ANLY 510 Kao Principals and Applications\\Lecture and other materials")
library(readxl)## Warning: package 'readxl' was built under R version 3.6.3data4 <- read_excel("Lecture 4 ANOVAExample2.xlsx")
names(data4)## [1] "Valuation" "ProductorGamble" "Condition" "Session"str(data4)## Classes 'tbl_df', 'tbl' and 'data.frame': 240 obs. of 4 variables:
## $ Valuation : num 7 0 2 10 6 5 13 10 4 9 ...
## $ ProductorGamble: chr "Product" "Gamble" "Product" "Gamble" ...
## $ Condition : chr "Chooser" "Chooser" "Chooser" "Chooser" ...
## $ Session : num 1 2 3 1 2 3 1 2 3 1 ...summary(data4)## Valuation ProductorGamble Condition Session
## Min. : 0.000 Length:240 Length:240 Min. :1
## 1st Qu.: 4.000 Class :character Class :character 1st Qu.:1
## Median : 6.000 Mode :character Mode :character Median :2
## Mean : 6.229 Mean :2
## 3rd Qu.: 9.000 3rd Qu.:3
## Max. :16.000 Max. :3Assumptions
library(moments)
plot(density(data4$Valuation), main = "Density Plot")
qqnorm(data4$Valuation)
agostino.test(data4$Valuation) # D'Agostino skewness test##
## D'Agostino skewness test
##
## data: data4$Valuation
## skew = 0.10977, z = 0.71253, p-value = 0.4761
## alternative hypothesis: data have a skewnessshapiro.test(data4$Valuation) # Shapiro-Wilk normality test##
## Shapiro-Wilk normality test
##
## data: data4$Valuation
## W = 0.97406, p-value = 0.0002225anscombe.test(data4$Valuation) # Anscombe-Glynn kurtosis test##
## Anscombe-Glynn kurtosis test
##
## data: data4$Valuation
## kurt = 2.3083, z = -3.2073, p-value = 0.00134
## alternative hypothesis: kurtosis is not equal to 3#Residual plot (lm - linear model)
valuation.lm <- lm(Valuation ~ Condition, data = data4)
valuation.lm##
## Call:
## lm(formula = Valuation ~ Condition, data = data4)
##
## Coefficients:
## (Intercept) ConditionChooser ConditionSeller
## 4.400 2.150 3.338summary(valuation.lm)##
## Call:
## lm(formula = Valuation ~ Condition, data = data4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.737 -2.438 -0.400 2.487 8.600
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.4000 0.3820 11.519 < 2e-16 ***
## ConditionChooser 2.1500 0.5402 3.980 9.17e-05 ***
## ConditionSeller 3.3375 0.5402 6.178 2.80e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.417 on 237 degrees of freedom
## Multiple R-squared: 0.142, Adjusted R-squared: 0.1348
## F-statistic: 19.61 on 2 and 237 DF, p-value: 1.31e-08valuation.res <- resid(valuation.lm) #OR valuation.res <- residuals(valuation.lm)
valuation.res## 1 2 3 4 5 6 7 8 9 10
## 0.4500 -6.5500 -4.5500 3.4500 -0.5500 -1.5500 6.4500 3.4500 -2.5500 2.4500
## 11 12 13 14 15 16 17 18 19 20
## 3.4500 3.4500 -1.5500 -6.5500 4.4500 3.4500 -5.5500 2.4500 -3.5500 -6.5500
## 21 22 23 24 25 26 27 28 29 30
## -0.5500 -1.5500 -1.5500 -1.5500 -0.5500 -4.5500 5.4500 -5.5500 1.4500 -1.5500
## 31 32 33 34 35 36 37 38 39 40
## 5.4500 -5.5500 -2.5500 -6.5500 -0.5500 -1.5500 4.4500 0.4500 -0.5500 2.4500
## 41 42 43 44 45 46 47 48 49 50
## 0.4500 -2.5500 5.4500 -0.5500 5.4500 0.4500 1.4500 0.4500 0.4500 -0.5500
## 51 52 53 54 55 56 57 58 59 60
## 3.4500 -0.5500 3.4500 1.4500 -3.5500 4.4500 -0.5500 -0.5500 1.4500 2.4500
## 61 62 63 64 65 66 67 68 69 70
## -4.5500 -5.5500 0.4500 3.4500 0.4500 4.4500 1.4500 -2.5500 -1.5500 0.4500
## 71 72 73 74 75 76 77 78 79 80
## 0.4500 0.4500 -0.5500 1.4500 -0.5500 -2.5500 5.4500 2.4500 2.4500 -6.5500
## 81 82 83 84 85 86 87 88 89 90
## -0.4000 -2.4000 -3.4000 -2.4000 1.6000 -2.4000 -0.4000 -1.4000 1.6000 -4.4000
## 91 92 93 94 95 96 97 98 99 100
## 2.6000 -2.4000 1.6000 -2.4000 -2.4000 -2.4000 3.6000 -4.4000 -0.4000 -0.4000
## 101 102 103 104 105 106 107 108 109 110
## 3.6000 -2.4000 4.6000 -2.4000 2.6000 -2.4000 -1.4000 -4.4000 5.6000 -4.4000
## 111 112 113 114 115 116 117 118 119 120
## 2.6000 -2.4000 5.6000 -4.4000 1.6000 3.6000 -0.4000 -3.4000 -2.4000 -2.4000
## 121 122 123 124 125 126 127 128 129 130
## 6.6000 -4.4000 -4.4000 0.6000 1.6000 2.6000 -1.4000 -4.4000 0.6000 2.6000
## 131 132 133 134 135 136 137 138 139 140
## 2.6000 -0.4000 5.6000 0.6000 0.6000 -0.4000 5.6000 -4.4000 -4.4000 -2.4000
## 141 142 143 144 145 146 147 148 149 150
## 8.6000 3.6000 3.6000 -1.4000 -0.4000 1.6000 4.6000 -3.4000 -0.4000 -2.4000
## 151 152 153 154 155 156 157 158 159 160
## -1.4000 -3.4000 5.6000 4.6000 1.6000 0.6000 -3.4000 4.6000 2.6000 -4.4000
## 161 162 163 164 165 166 167 168 169 170
## 2.2625 -7.7375 1.2625 -1.7375 -2.7375 -0.7375 1.2625 -5.7375 4.2625 -1.7375
## 171 172 173 174 175 176 177 178 179 180
## 3.2625 -5.7375 -3.7375 2.2625 -0.7375 -3.7375 2.2625 2.2625 1.2625 0.2625
## 181 182 183 184 185 186 187 188 189 190
## 3.2625 -3.7375 0.2625 -4.7375 -0.7375 -2.7375 2.2625 -2.7375 7.2625 -1.7375
## 191 192 193 194 195 196 197 198 199 200
## 3.2625 -1.7375 -2.7375 -3.7375 1.2625 3.2625 -3.7375 -5.7375 2.2625 -4.7375
## 201 202 203 204 205 206 207 208 209 210
## 1.2625 0.2625 5.2625 -1.7375 5.2625 -1.7375 1.2625 -1.7375 1.2625 -6.7375
## 211 212 213 214 215 216 217 218 219 220
## -1.7375 -6.7375 8.2625 0.2625 2.2625 3.2625 6.2625 2.2625 7.2625 3.2625
## 221 222 223 224 225 226 227 228 229 230
## -3.7375 3.2625 5.2625 -2.7375 5.2625 -2.7375 -2.7375 -2.7375 0.2625 -2.7375
## 231 232 233 234 235 236 237 238 239 240
## 1.2625 2.2625 3.2625 -3.7375 4.2625 0.2625 -1.7375 -0.7375 3.2625 -2.7375plot(data4$Valuation, valuation.res, ylab = "Residual", xlab = "Condition", main = "Independence of Observation")
abline(0, 0)
bartlett.test(data4$Valuation , data4$Condition)##
## Bartlett test of homogeneity of variances
##
## data: data4$Valuation and data4$Condition
## Bartlett's K-squared = 0.57097, df = 2, p-value = 0.7516# Variance are equal. p-value = 0.7516 failed to reject the null hypothesis
#Checking variance among groups
tapply(data4$Valuation , data4$Condition, var)## Buyer Chooser Seller
## 10.87595 11.33924 12.80364#ratio of largest variance to smallest variance 12.80/10.87 = 1.17 - is way less than 3 (signifies that there is no issue with failing to reject null hypothesis)#After checking all the assumptions fulfill our criteria, now we will perform ANOVA with Blocking Design
#IV - ProductorGamble Condition Session (predictor) categorical
#DV - valuation (outcome) continuous
#Testing Additivity of Interactions - To check there is no interaction between predictor/IV - conditions and block - age (Interaction variable - factor(Condition)*factor(Age))
#Perform the linear model:
model1 <- aov(Valuation ~ Condition*ProductorGamble, data = data4) #we use first predictor*second predictor to test (add) the most important interaction effect. If we have significant interaction effect, we consider the interaction effect and go back to check the main effect.
model1## Call:
## aov(formula = Valuation ~ Condition * ProductorGamble, data = data4)
##
## Terms:
## Condition ProductorGamble Condition:ProductorGamble Residuals
## Sum of Squares 457.9083 429.3375 52.6750 2284.4750
## Deg. of Freedom 2 1 2 234
##
## Residual standard error: 3.124534
## Estimated effects may be unbalancedsummary(model1) ## Df Sum Sq Mean Sq F value Pr(>F)
## Condition 2 457.9 229.0 23.452 5.21e-10 ***
## ProductorGamble 1 429.3 429.3 43.977 2.28e-10 ***
## Condition:ProductorGamble 2 52.7 26.3 2.698 0.0695 .
## Residuals 234 2284.5 9.8
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Interaction effect - looks almost significant at p-value < 0.1 level but not at < 0.05 level
#lets check the main effects - condition and productorgamble
model2 <- aov(Valuation ~ Condition, data = data4)
model2## Call:
## aov(formula = Valuation ~ Condition, data = data4)
##
## Terms:
## Condition Residuals
## Sum of Squares 457.9083 2766.4875
## Deg. of Freedom 2 237
##
## Residual standard error: 3.416569
## Estimated effects may be unbalancedsummary(model2)## Df Sum Sq Mean Sq F value Pr(>F)
## Condition 2 457.9 228.95 19.61 1.31e-08 ***
## Residuals 237 2766.5 11.67
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Effect of Condition is signifiacnt
#OTHER MOdels FOR COmparison
model3 <- aov(Valuation ~ ProductorGamble, data = data4)
model3## Call:
## aov(formula = Valuation ~ ProductorGamble, data = data4)
##
## Terms:
## ProductorGamble Residuals
## Sum of Squares 429.3375 2795.0583
## Deg. of Freedom 1 238
##
## Residual standard error: 3.426944
## Estimated effects may be unbalancedsummary(model3)## Df Sum Sq Mean Sq F value Pr(>F)
## ProductorGamble 1 429.3 429.3 36.56 5.69e-09 ***
## Residuals 238 2795.1 11.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Effect of ProductorGamble is also signifiacnt
#Now that we know both of our predictors are significant, we put them together in our final model
model4 <- aov(Valuation ~ ProductorGamble + Condition, data = data4)
model4## Call:
## aov(formula = Valuation ~ ProductorGamble + Condition, data = data4)
##
## Terms:
## ProductorGamble Condition Residuals
## Sum of Squares 429.3375 457.9083 2337.1500
## Deg. of Freedom 1 2 236
##
## Residual standard error: 3.146932
## Estimated effects may be unbalancedsummary(model4)## Df Sum Sq Mean Sq F value Pr(>F)
## ProductorGamble 1 429.3 429.3 43.35 2.93e-10 ***
## Condition 2 457.9 229.0 23.12 6.77e-10 ***
## Residuals 236 2337.1 9.9
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Using Chi Sq Test to check the difference between each model and we use the residual part to decide which model is better
comparison1 <- anova(model2, model3)
comparison1 #no significant difference (p-value > 0.05)## Analysis of Variance Table
##
## Model 1: Valuation ~ Condition
## Model 2: Valuation ~ ProductorGamble
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 237 2766.5
## 2 238 2795.1 -1 -28.571 2.4476 0.119#results suggest no significant differenc between model 2 and 3
comparison2 <- anova(model3, model4)
comparison2 #significant difference nted (p-value < 0.05)## Analysis of Variance Table
##
## Model 1: Valuation ~ ProductorGamble
## Model 2: Valuation ~ ProductorGamble + Condition
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 238 2795.1
## 2 236 2337.2 2 457.91 23.119 6.775e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#results suggest significant differenc between model 3 and 4
#SO we should include both the predictors productorgamble and conditions in out modelgraph <- interaction.plot(data4$Condition, data4$ProductorGamble, data4$Valuation)
graph #demonstrate not significant interaction effect## NULL#To find out what group differs from the others
library(pgirmess)## Warning: package 'pgirmess' was built under R version 3.6.3#Tukey’s Test: compare all the potential combination chooser at gamble level with buyer at gamble level
TukeyHSD(model2) #for two way/factorial anova (predictors - Condition and ProductorGamble)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Valuation ~ Condition, data = data4)
##
## $Condition
## diff lwr upr p adj
## Chooser-Buyer 2.1500 0.87590343 3.424097 0.0002695
## Seller-Buyer 3.3375 2.06340343 4.611597 0.0000000
## Seller-Chooser 1.1875 -0.08659657 2.461597 0.0735348TukeyHSD(model1) #for one way anova (predictor - Condition)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Valuation ~ Condition * ProductorGamble, data = data4)
##
## $Condition
## diff lwr upr p adj
## Chooser-Buyer 2.1500 0.9847137 3.315286 0.0000598
## Seller-Buyer 3.3375 2.1722137 4.502786 0.0000000
## Seller-Chooser 1.1875 0.0222137 2.352786 0.0446738
##
## $ProductorGamble
## diff lwr upr p adj
## Product-Gamble 2.675 1.880288 3.469712 0
##
## $`Condition:ProductorGamble`
## diff lwr upr p adj
## Chooser:Gamble-Buyer:Gamble 2.950 0.9423887 4.957611 0.0004916
## Seller:Gamble-Buyer:Gamble 3.025 1.0173887 5.032611 0.0003167
## Buyer:Product-Buyer:Gamble 3.000 0.9923887 5.007611 0.0003671
## Chooser:Product-Buyer:Gamble 4.350 2.3423887 6.357611 0.0000000
## Seller:Product-Buyer:Gamble 6.650 4.6423887 8.657611 0.0000000
## Seller:Gamble-Chooser:Gamble 0.075 -1.9326113 2.082611 0.9999980
## Buyer:Product-Chooser:Gamble 0.050 -1.9576113 2.057611 0.9999997
## Chooser:Product-Chooser:Gamble 1.400 -0.6076113 3.407611 0.3432708
## Seller:Product-Chooser:Gamble 3.700 1.6923887 5.707611 0.0000040
## Buyer:Product-Seller:Gamble -0.025 -2.0326113 1.982611 1.0000000
## Chooser:Product-Seller:Gamble 1.325 -0.6826113 3.332611 0.4069019
## Seller:Product-Seller:Gamble 3.625 1.6173887 5.632611 0.0000068
## Chooser:Product-Buyer:Product 1.350 -0.6576113 3.357611 0.3851503
## Seller:Product-Buyer:Product 3.650 1.6423887 5.657611 0.0000057
## Seller:Product-Chooser:Product 2.300 0.2923887 4.307611 0.0144395library(pastecs)## Warning: package 'pastecs' was built under R version 3.6.3library(compute.es)## Warning: package 'compute.es' was built under R version 3.6.3#First get the relevant stats for each factor level (only relevant output pasted below):
by(data4$Valuation, data4$Condition, stat.desc)## data4$Condition: Buyer
## nbr.val nbr.null nbr.na min max range
## 80.0000000 11.0000000 0.0000000 0.0000000 13.0000000 13.0000000
## sum median mean SE.mean CI.mean.0.95 var
## 352.0000000 4.0000000 4.4000000 0.3687131 0.7339051 10.8759494
## std.dev coef.var
## 3.2978704 0.7495160
## ------------------------------------------------------------
## data4$Condition: Chooser
## nbr.val nbr.null nbr.na min max range
## 80.0000000 5.0000000 0.0000000 0.0000000 13.0000000 13.0000000
## sum median mean SE.mean CI.mean.0.95 var
## 524.0000000 7.0000000 6.5500000 0.3764844 0.7493735 11.3392405
## std.dev coef.var
## 3.3673789 0.5141036
## ------------------------------------------------------------
## data4$Condition: Seller
## nbr.val nbr.null nbr.na min max range
## 80.0000000 1.0000000 0.0000000 0.0000000 16.0000000 16.0000000
## sum median mean SE.mean CI.mean.0.95 var
## 619.0000000 8.0000000 7.7375000 0.4000569 0.7962933 12.8036392
## std.dev coef.var
## 3.5782173 0.4624514#n - sample size, M - mean, SD - standard deviation
#Buyer (n = 80, M = 4.40 , SD = 3.29)
#Chooser (n = 80, M = 6.55, SD = 3.36)
#Seller (n = 80, M = 7.73, SD = 3.57)
#Use these values to compute a few standard Measures of Effect Size (MES or mes) for any pair of interest
#OPtion 1 will compare buyer vs chooser ~ valuation vs Condition
mes(6.55, 4.40, 3.36, 3.29, 80, 80)## Mean Differences ES:
##
## d [ 95 %CI] = 0.65 [ 0.33 , 0.96 ]
## var(d) = 0.03
## p-value(d) = 0
## U3(d) = 74.1 %
## CLES(d) = 67.62 %
## Cliff's Delta = 0.35
##
## g [ 95 %CI] = 0.64 [ 0.33 , 0.96 ]
## var(g) = 0.03
## p-value(g) = 0
## U3(g) = 74.01 %
## CLES(g) = 67.55 %
##
## Correlation ES:
##
## r [ 95 %CI] = 0.31 [ 0.16 , 0.44 ]
## var(r) = 0
## p-value(r) = 0
##
## z [ 95 %CI] = 0.32 [ 0.16 , 0.48 ]
## var(z) = 0.01
## p-value(z) = 0
##
## Odds Ratio ES:
##
## OR [ 95 %CI] = 3.23 [ 1.82 , 5.75 ]
## p-value(OR) = 0
##
## Log OR [ 95 %CI] = 1.17 [ 0.6 , 1.75 ]
## var(lOR) = 0.09
## p-value(Log OR) = 0
##
## Other:
##
## NNT = 4.49
## Total N = 160#OPtion 2 will compare choser vs seller ~ valuation vs Condition
mes(7.73, 6.55, 3.57, 3.36, 80, 80)## Mean Differences ES:
##
## d [ 95 %CI] = 0.34 [ 0.03 , 0.65 ]
## var(d) = 0.03
## p-value(d) = 0.03
## U3(d) = 63.32 %
## CLES(d) = 59.51 %
## Cliff's Delta = 0.19
##
## g [ 95 %CI] = 0.34 [ 0.03 , 0.65 ]
## var(g) = 0.03
## p-value(g) = 0.03
## U3(g) = 63.26 %
## CLES(g) = 59.47 %
##
## Correlation ES:
##
## r [ 95 %CI] = 0.17 [ 0.01 , 0.32 ]
## var(r) = 0.01
## p-value(r) = 0.03
##
## z [ 95 %CI] = 0.17 [ 0.01 , 0.33 ]
## var(z) = 0.01
## p-value(z) = 0.03
##
## Odds Ratio ES:
##
## OR [ 95 %CI] = 1.85 [ 1.05 , 3.27 ]
## p-value(OR) = 0.03
##
## Log OR [ 95 %CI] = 0.62 [ 0.05 , 1.18 ]
## var(lOR) = 0.08
## p-value(Log OR) = 0.03
##
## Other:
##
## NNT = 9.25
## Total N = 160#OPtion 3 will compare buyer vs seller ~ valuation vs Condition
mes(7.73, 4.40, 3.57, 3.29, 80, 80)## Mean Differences ES:
##
## d [ 95 %CI] = 0.97 [ 0.64 , 1.3 ]
## var(d) = 0.03
## p-value(d) = 0
## U3(d) = 83.4 %
## CLES(d) = 75.36 %
## Cliff's Delta = 0.51
##
## g [ 95 %CI] = 0.97 [ 0.64 , 1.29 ]
## var(g) = 0.03
## p-value(g) = 0
## U3(g) = 83.28 %
## CLES(g) = 75.26 %
##
## Correlation ES:
##
## r [ 95 %CI] = 0.44 [ 0.3 , 0.56 ]
## var(r) = 0
## p-value(r) = 0
##
## z [ 95 %CI] = 0.47 [ 0.31 , 0.63 ]
## var(z) = 0.01
## p-value(z) = 0
##
## Odds Ratio ES:
##
## OR [ 95 %CI] = 5.81 [ 3.21 , 10.52 ]
## p-value(OR) = 0
##
## Log OR [ 95 %CI] = 1.76 [ 1.17 , 2.35 ]
## var(lOR) = 0.09
## p-value(Log OR) = 0
##
## Other:
##
## NNT = 2.85
## Total N = 160#Effect size (more than or equal to) < = 0.1 is small, 0.25 is medium, 0.4 is large (Cohen, 1988)#To check if different session (eg- Mon, Tue, Wed, etc or am, pm or weekday, weekend) acts as a confounding factor and ALSO do we need to BLOCK for the INTERACTION EFFECT -
#Test Additivity of Interaction - check if the block (session) has any interaction with our predictors (productorgamble and condition)
model5 <- aov(Valuation ~ Condition*ProductorGamble*factor(Session), data = data4)
model5## Call:
## aov(formula = Valuation ~ Condition * ProductorGamble * factor(Session),
## data = data4)
##
## Terms:
## Condition ProductorGamble factor(Session)
## Sum of Squares 457.9083 429.3375 63.1434
## Deg. of Freedom 2 1 2
## Condition:ProductorGamble Condition:factor(Session)
## Sum of Squares 55.2222 23.6435
## Deg. of Freedom 2 4
## ProductorGamble:factor(Session)
## Sum of Squares 12.0510
## Deg. of Freedom 2
## Condition:ProductorGamble:factor(Session) Residuals
## Sum of Squares 56.1998 2126.8901
## Deg. of Freedom 4 222
##
## Residual standard error: 3.095252
## Estimated effects may be unbalancedsummary(model5)## Df Sum Sq Mean Sq F value Pr(>F)
## Condition 2 457.9 229.0 23.898 3.98e-10
## ProductorGamble 1 429.3 429.3 44.813 1.75e-10
## factor(Session) 2 63.1 31.6 3.295 0.0389
## Condition:ProductorGamble 2 55.2 27.6 2.882 0.0581
## Condition:factor(Session) 4 23.6 5.9 0.617 0.6509
## ProductorGamble:factor(Session) 2 12.1 6.0 0.629 0.5341
## Condition:ProductorGamble:factor(Session) 4 56.2 14.0 1.467 0.2133
## Residuals 222 2126.9 9.6
##
## Condition ***
## ProductorGamble ***
## factor(Session) *
## Condition:ProductorGamble .
## Condition:factor(Session)
## ProductorGamble:factor(Session)
## Condition:ProductorGamble:factor(Session)
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#results suggest there is no significant interactin effect (Condition:ProductorGamble:factor(Session) p-value > 0.05)
#So, Session can be treated as a BLOCK
#Testing if predictors (condition and productorgamble) has an interaction effect with session added as a Block
model6 <- aov(Valuation ~ Condition*ProductorGamble + factor(Session), data = data4)
model6## Call:
## aov(formula = Valuation ~ Condition * ProductorGamble + factor(Session),
## data = data4)
##
## Terms:
## Condition ProductorGamble factor(Session)
## Sum of Squares 457.9083 429.3375 63.1434
## Deg. of Freedom 2 1 2
## Condition:ProductorGamble Residuals
## Sum of Squares 55.2222 2218.7844
## Deg. of Freedom 2 232
##
## Residual standard error: 3.092527
## Estimated effects may be unbalancedsummary(model6)## Df Sum Sq Mean Sq F value Pr(>F)
## Condition 2 457.9 229.0 23.940 3.53e-10 ***
## ProductorGamble 1 429.3 429.3 44.892 1.56e-10 ***
## factor(Session) 2 63.1 31.6 3.301 0.0386 *
## Condition:ProductorGamble 2 55.2 27.6 2.887 0.0577 .
## Residuals 232 2218.8 9.6
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Not significant result noted#To find out what group differs from the others
library(pgirmess)
#Tukey’s Test: compare all the potential combination chooser at gamble level with buyer at gamble level
TukeyHSD(model6) #for two way/factorial anova (predictors - Condition and ProductorGamble with session as a block)## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = Valuation ~ Condition * ProductorGamble + factor(Session), data = data4)
##
## $Condition
## diff lwr upr p adj
## Chooser-Buyer 2.1500 0.99658685 3.303413 0.0000496
## Seller-Buyer 3.3375 2.18408685 4.490913 0.0000000
## Seller-Chooser 1.1875 0.03408685 2.340913 0.0419550
##
## $ProductorGamble
## diff lwr upr p adj
## Product-Gamble 2.675 1.888394 3.461606 0
##
## $`factor(Session)`
## diff lwr upr p adj
## 2-1 -1.2292188 -2.3826319 -0.0758056 0.0336355
## 3-1 -0.3898438 -1.5432569 0.7635694 0.7050740
## 3-2 0.8393750 -0.3140381 1.9927881 0.2011292
##
## $`Condition:ProductorGamble`
## diff lwr upr p adj
## Chooser:Gamble-Buyer:Gamble 2.970241444 0.9830530 4.957430 0.0003661
## Seller:Gamble-Buyer:Gamble 3.019379981 1.0321915 5.006568 0.0002729
## Buyer:Product-Buyer:Gamble 3.009747617 1.0225591 4.996936 0.0002891
## Chooser:Product-Buyer:Gamble 4.339506173 2.3523177 6.326695 0.0000000
## Seller:Product-Buyer:Gamble 6.665367636 4.6781792 8.652556 0.0000000
## Seller:Gamble-Chooser:Gamble 0.049138537 -1.9380499 2.036327 0.9999997
## Buyer:Product-Chooser:Gamble 0.039506173 -1.9476823 2.026695 0.9999999
## Chooser:Product-Chooser:Gamble 1.369264729 -0.6179237 3.356453 0.3569160
## Seller:Product-Chooser:Gamble 3.695126192 1.7079377 5.682315 0.0000032
## Buyer:Product-Seller:Gamble -0.009632364 -1.9968208 1.977556 1.0000000
## Chooser:Product-Seller:Gamble 1.320126192 -0.6670623 3.307315 0.3992214
## Seller:Product-Seller:Gamble 3.645987654 1.6587992 5.633176 0.0000046
## Chooser:Product-Buyer:Product 1.329758556 -0.6574299 3.316947 0.3907629
## Seller:Product-Buyer:Product 3.655620019 1.6684315 5.642808 0.0000043
## Seller:Product-Chooser:Product 2.325861463 0.3386730 4.313050 0.0114799Summary Write Up
A study was conducted to investigate whether, taken together, a person’s status and product gamble form affected individuals’ valuation of the product. Observations from the study were analyzed by conducting a two-way analysis of variance with the two independent variables (condition and product or gamble) and the dependent variable (valuation) using R version 3.61. First, all assumptions are met and there is no adjustment made. ANOVA analysis revealed there is no statistically significant interaction effect (F(2, 234) = 2.698, p = .070). Results also suggest that the valuations of an item’s worth was affected by the status of (as buyers, sellers, or choosers) the participants (F(2, 234) = 23.45, p < .001) and product or gamble (F(1, 234) = 43.98, p < .001).